Calibration of a multifactor model for the forward markets of several commodities

Abstract

We propose a model for the evolution of forward prices of several commodities, which is an extension of the factor forward model in [1, 2], to a market where multiple commodities are traded. We calibrate this model in a market where forward contracts on multiple commodities are present, using historical forward prices. First, we calibrate separately the four coefficients of each individual commodity, using an approach based on quadratic variation/covariation of forward prices. Then, with the same technique, we pass to the estimation of the mutual correlation among the Brownian motions driving the different commodities. This calibration is compared to a calibration method used by practitioners, which uses rolling time series and requires a modification of the model, but turns out to be more accurate in practice, especially with a low frequency of observed transaction. We present efficient methods to perform the calibration with both methods, as well as the calibration of the intercommodity correlation matrix. Then we calibrate our model to WTI, ICE Brent and ICE Gasoil forward prices. Finally we present a method for estimating spot volatility from forward parameters, with an application to the WTI spot volatility.

Keywords:

• two-factor model for commodity forward prices
• historical calibration
• quadratic variation/covariation
• rolling time series
• non-convex optimization
• semidefinite programming

AMS Subject Classifications:

• 60H35
• 62H20
• 62M05
• 90C22
• 90C26
• 91B25

Acknowledgments

The authors wish to thank Fabio Antonelli, Alberto Del Pia, Paolo Foschi, Francesco Rinaldi, Wolfgang J. Runggaldier and all the participants to the Conference on Energy & Finance in Trondheim (Norway), October 4–5, 2012 and to the XIV Workshop in Quantitative Finance in Rimini (Italy), January 24–25, 2013, together with two anonymous referees, for various comments and suggestions.

Notes

1 In order to be rigorous, one should define as in the previous sections: we choose not to do so, as it would make the notation heavier and we believe that all is clear from the context.